Extended redundancy analysis (ERA) combines linear regression with dimension reduction to explore the directional relationships between multiple sets of predictors and outcome variables in a parsimonious manner. It aims to extract a component from each set of predictors in such a way that it accounts for the maximum variance of outcome variables. In this article, we extend ERA into the Bayesian framework, called Bayesian ERA (BERA). The advantages of BERA are threefold. First, BERA enables to make statistical inferences based on samples drawn from the joint posterior distribution of parameters obtained from a Markov chain Monte Carlo algorithm. As such, it does not necessitate any resampling method, which is on the other hand required for (frequentist’s) ordinary ERA to test the statistical significance of parameter estimates. Second, it formally incorporates relevant information obtained from previous research into analyses by specifying informative power prior distributions. Third, BERA handles missing data by implementing multiple imputation using a Markov Chain Monte Carlo algorithm, avoiding the potential bias of parameter estimates due to missing data. We assess the performance of BERA through simulation studies and apply BERA to real data regarding academic achievement.

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A multivariate reduced-rank growth curve model is proposed that extends the univariate reducedrank growth curve model to the multivariate case, in which several response variables are measured over multiple time points. The proposed model allows us to investigate the relationships among a number of response variables in a more parsimonious way than the traditional growth curve model. In addition, the method is more flexible than the traditional growth curve model. For example, response variables do not have to be measured at the same time points, nor the same number of time points. It is also possible to
apply various kinds of basis function matrices with different ranks across response variables. It is not necessary to specify an entire set of basis functions in advance. Examples are given for illustration.

The first example is part of the National Longitudinal Survey of Youth (NLSY), conducted by the U. S. Department of Labor. Starting in 1986, a large sample of children and their mothers were administered a set of assessment instruments every other year until 1992. Interviews were conducted on each child and her/his mother about the child. From the original sample of children and mothers, we analyzed a smaller sample of child-mother pairs, provided in Curran (1998). (Note that Curran (1998) considered only one biological child from each mother.) The sample consisted of 221 pairs of children and mothers, who completed interviews at four time points. Two variables were repeatedly measured over the four time points: Antisocial behavior and reading recognition. .... Besides the two repeatedly measured variables, three variables were assessed once at the initial time point: cognitive stimulation for children at home, emotional support for children at home, and gender.